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Mirrors > Home > MPE Home > Th. List > isowe | Structured version Visualization version GIF version |
Description: An isomorphism preserves the property of being a well-ordering. Proposition 6.32(3) of [TakeutiZaring] p. 33. (Contributed by NM, 30-Apr-2004.) (Revised by Mario Carneiro, 18-Nov-2014.) |
Ref | Expression |
---|---|
isowe | ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) |
Step | Hyp | Ref | Expression |
---|---|---|---|
1 | isofr 7345 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Fr 𝐴 ↔ 𝑆 Fr 𝐵)) | |
2 | isoso 7351 | . . 3 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 Or 𝐴 ↔ 𝑆 Or 𝐵)) | |
3 | 1, 2 | anbi12d 631 | . 2 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → ((𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴) ↔ (𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵))) |
4 | df-we 5630 | . 2 ⊢ (𝑅 We 𝐴 ↔ (𝑅 Fr 𝐴 ∧ 𝑅 Or 𝐴)) | |
5 | df-we 5630 | . 2 ⊢ (𝑆 We 𝐵 ↔ (𝑆 Fr 𝐵 ∧ 𝑆 Or 𝐵)) | |
6 | 3, 4, 5 | 3bitr4g 314 | 1 ⊢ (𝐻 Isom 𝑅, 𝑆 (𝐴, 𝐵) → (𝑅 We 𝐴 ↔ 𝑆 We 𝐵)) |
Colors of variables: wff setvar class |
Syntax hints: → wi 4 ↔ wb 205 ∧ wa 395 Or wor 5584 Fr wfr 5625 We wwe 5627 Isom wiso 6544 |
This theorem was proved from axioms: ax-mp 5 ax-1 6 ax-2 7 ax-3 8 ax-gen 1790 ax-4 1804 ax-5 1906 ax-6 1964 ax-7 2004 ax-8 2101 ax-9 2109 ax-10 2130 ax-11 2147 ax-12 2167 ax-ext 2699 ax-rep 5280 ax-sep 5294 ax-nul 5301 ax-pr 5424 |
This theorem depends on definitions: df-bi 206 df-an 396 df-or 847 df-3or 1086 df-3an 1087 df-tru 1537 df-fal 1547 df-ex 1775 df-nf 1779 df-sb 2061 df-mo 2530 df-eu 2559 df-clab 2706 df-cleq 2720 df-clel 2806 df-ne 2937 df-ral 3058 df-rex 3067 df-rab 3429 df-v 3472 df-dif 3948 df-un 3950 df-in 3952 df-ss 3962 df-nul 4320 df-if 4526 df-sn 4626 df-pr 4628 df-op 4632 df-uni 4905 df-br 5144 df-opab 5206 df-id 5571 df-po 5585 df-so 5586 df-fr 5628 df-we 5630 df-xp 5679 df-rel 5680 df-cnv 5681 df-co 5682 df-dm 5683 df-rn 5684 df-res 5685 df-ima 5686 df-iota 6495 df-fun 6545 df-fn 6546 df-f 6547 df-f1 6548 df-fo 6549 df-f1o 6550 df-fv 6551 df-isom 6552 |
This theorem is referenced by: f1owe 7356 hartogslem1 9560 oemapwe 9712 om2uzoi 13947 om2noseqoi 28170 |
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